(CNN) -- The Williams sisters cruised into the second round of the U.S. Open but their fellow former world No. 1 Caroline Wozniacki became the highest-profile casualty of the women's draw in New York.

Wimbledon and Olympic champion Serena Williams defied windy conditions to thrash fellow American Coco Vandeweghe 6-1 6-1 in just 55 minutes on Tuesday to set up a match against experienced Spaniard Maria Jose Martinez Sanchez.

"The match was so weird, the conditions were so tough, I couldn't really play my game. She couldn't really play her game. It's not the best opportunity to grade yourself," said fourth seed Williams, who also beat 2008 U.S. Open girls' champion Vandeweghe in the final of the Stanford tournament in July before heading back to Wimbledon for London 2012.

Serena won the U.S. Open for the third time in 2008, and older sibling Venus is seeking to match that feat.

The 32-year-old, unseeded after falling to 46th in the rankings after being diagnosed an auto-immune disease this time last year, began her bid by beating compatriot Bethanie Mattek-Sands 6-3 6-1.

"There comes a point when you have to start realizing you can't play through everything," said Williams, who next faces a tough test in the form of German sixth seed Angelique Kerber.

"For me that's been just about a year, year and a half that I've finally gotten smart."

Wimbledon semifinalist Kerber, who beat Venus in Madrid and then at the Olympics, thrashed Britain's Anne Keothavong 6-2 6-0.

"I have to put a lot of balls in the court and execute my game well if I want to win against her," Williams said.

Wozniacki has dropped from the top of the rankings to ninth since last year's U.S. Open, and the Dane's hopes of winning a first grand slam title seem further away than ever.

She reached the New York final in 2009, but on Tuesday lost 6-2 6-2 to 96th-ranked Romanian Irina-Camelia Begu to suffer her second successive first-round exit at a grand slam.

"I think it's a temporary issue," said the 22-year-old, who played with a strapped knee after pulling out during hersemifinal at last week's New Haven tournament.

"Hopefully it will get better quickly. It's frustrating to have some injuries, but it happens to everyone. It's just about moving on."

Begu, who turned 22 on Sunday, was the WTA Tour's newcomer of the year in 2011 but has dropped from a career-high ranking of 38th.

She will next play 58th-ranked Spaniard Silvia Soler-Espinosa, who reached the third round last year.

Former French Open champion Francesca Schiavone was the other big name to depart on Tuesday as the Italian 22nd seed lost to American teenager Sloane Stephens -- touted by many as the natural successor to the Williams sisters.

Stephens will next play 141st-ranked German Tatjana Malek, who reached the second round in New York for the first time.

The 19-year-old has surged up the rankings to 44th, building on her third-round achievement at last year's U.S. Open with a similar showing at Wimbledon and a last-16 appearance at Roland Garros.

The Wimbledon finalist has lost in the second round at Flushing Meadows for the past three years, and this time faces unseeded Spaniard Carla Suarez Navarro -- who reached the last 16 in 2011.

"I'm just very happy that I could give it my best in the first match. It's always difficult for the first match," said the 23-year-old, whose younger sister Urzsula lost to Italian 20th seed Roberta Vinci.

"It's always a bit harder because the people expect you to play well and go far in the tournament. But I'm just trying not to really think about pressure and just do my best."

Former world No. 1 Ana Ivanovic beat Ukrainian Elina Svitolina 6-3 6-3 to set up a clash with Sweden's Sofia Arvidsson.

The Serbian 12th seed has never been past the fourth round in New York, her worst showing at the four grand slams.

"My big goal is to break the fourth round and get into quarterfinals. Obviously that's something I'm working hard. It's not easy, but I really tried to focus on taking it one match at a time, because sometimes when you get overexcited it doesn't really work for you the way you hope for," she said.